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Mirrors > Home > MPE Home > Th. List > ixxss1 | Structured version Visualization version GIF version |
Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
ixx.1 | ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) |
ixxss1.2 | ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)}) |
ixxss1.3 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤)) |
Ref | Expression |
---|---|
ixxss1 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ixxss1.2 | . . . . . . . 8 ⊢ 𝑃 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)}) | |
2 | 1 | elixx3g 12953 | . . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) ↔ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) ∧ (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶))) |
3 | 2 | simplbi 501 | . . . . . 6 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*)) |
4 | 3 | adantl 485 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*)) |
5 | 4 | simp3d 1146 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ ℝ*) |
6 | simplr 769 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑊𝐵) | |
7 | 2 | simprbi 500 | . . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶)) |
8 | 7 | adantl 485 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶)) |
9 | 8 | simpld 498 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵𝑇𝑤) |
10 | simpll 767 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴 ∈ ℝ*) | |
11 | 4 | simp1d 1144 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵 ∈ ℝ*) |
12 | ixxss1.3 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤)) | |
13 | 10, 11, 5, 12 | syl3anc 1373 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤)) |
14 | 6, 9, 13 | mp2and 699 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑅𝑤) |
15 | 8 | simprd 499 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤𝑆𝐶) |
16 | 4 | simp2d 1145 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐶 ∈ ℝ*) |
17 | ixx.1 | . . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)}) | |
18 | 17 | elixx1 12949 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶))) |
19 | 10, 16, 18 | syl2anc 587 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶))) |
20 | 5, 14, 15, 19 | mpbir3and 1344 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ (𝐴𝑂𝐶)) |
21 | 20 | ex 416 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝑤 ∈ (𝐵𝑃𝐶) → 𝑤 ∈ (𝐴𝑂𝐶))) |
22 | 21 | ssrdv 3912 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 {crab 3065 ⊆ wss 3871 class class class wbr 5058 (class class class)co 7218 ∈ cmpo 7220 ℝ*cxr 10871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5197 ax-nul 5204 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-resscn 10791 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-op 4553 df-uni 4825 df-iun 4911 df-br 5059 df-opab 5121 df-mpt 5141 df-id 5460 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-fv 6393 df-ov 7221 df-oprab 7222 df-mpo 7223 df-1st 7766 df-2nd 7767 df-xr 10876 |
This theorem is referenced by: iooss1 12975 limsupgord 15038 pnfnei 22122 dvfsumrlimge0 24932 dvfsumrlim2 24934 tanord1 25431 rlimcnp 25853 rlimcnp2 25854 dchrisum0lem2a 26403 pntleml 26497 pnt 26500 liminfgord 42978 |
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